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Hyperelliptic curve cryptography

Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insofar as the Jacobian of a hyperelliptic curve is an abelian group in which to do arithmetic, just as we use the group of points on an elliptic curve in ECC.



An (imaginary) hyperelliptic curve of genus   over a field   is given by the equation   where   is a polynomial of degree not larger than   and   is a monic polynomial of degree  . From this definition it follows that elliptic curves are hyperelliptic curves of genus 1. In hyperelliptic curve cryptography   is often a finite field. The Jacobian of  , denoted  , is a quotient group, thus the elements of the Jacobian are not points, they are equivalence classes of divisors of degree 0 under the relation of linear equivalence. This agrees with the elliptic curve case, because it can be shown that the Jacobian of an elliptic curve is isomorphic with the group of points on the elliptic curve.[1] The use of hyperelliptic curves in cryptography came about in 1989 from Neal Koblitz. Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA). The efficiency of implementing the arithmetic depends on the underlying finite field  , in practice it turns out that finite fields of characteristic 2 are a good choice for hardware implementations while software is usually faster in odd characteristic.[2]

The Jacobian on a hyperelliptic curve is an Abelian group and as such it can serve as group for the discrete logarithm problem (DLP). In short, suppose we have an Abelian group   and   an element of  , the DLP on   entails finding the integer   given two elements of  , namely   and  . The first type of group used was the multiplicative group of a finite field, later also Jacobians of (hyper)elliptic curves were used. If the hyperelliptic curve is chosen with care, then Pollard's rho method is the most efficient way to solve DLP. This means that, if the Jacobian has   elements, that the running time is exponential in  . This makes it possible to use Jacobians of a fairly small order, thus making the system more efficient. But if the hyperelliptic curve is chosen poorly, the DLP will become quite easy to solve. In this case there are known attacks which are more efficient than generic discrete logarithm solvers[3] or even subexponential.[4] Hence these hyperelliptic curves must be avoided. Considering various attacks on DLP, it is possible to list the features of hyperelliptic curves that should be avoided.

Attacks against the DLPEdit

All generic attacks on the discrete logarithm problem in finite abelian groups such as the Pohlig–Hellman algorithm and Pollard's rho method can be used to attack the DLP in the Jacobian of hyperelliptic curves. The Pohlig-Hellman attack reduces the difficulty of the DLP by looking at the order of the group we are working with. Suppose the group   that is used has   elements, where   is the prime factorization of  . Pohlig-Hellman reduces the DLP in   to DLPs in subgroups of order   for  . So for   the largest prime divisor of  , the DLP in   is just as hard to solve as the DLP in the subgroup of order  . Therefore we would like to choose   such that the largest prime divisor   of   is almost equal to   itself. Requiring   usually suffices.

The index calculus algorithm is another algorithm that can be used to solve DLP under some circumstances. For Jacobians of (hyper)elliptic curves there exists an index calculus attack on DLP. If the genus of the curve becomes too high, the attack will be more efficient than Pollard's rho. Today it is known that even a genus of   cannot assure security.[5] Hence we are left with elliptic curves and hyperelliptic curves of genus 2.

Another restriction on the hyperelliptic curves we can use comes from the Menezes-Okamoto-Vanstone-attack / Frey-Rück-attack. The first, often called MOV for short, was developed in 1993, the second came about in 1994. Consider a (hyper)elliptic curve   over a finite field   where   is the power of a prime number. Suppose the Jacobian of the curve has   elements and   is the largest prime divisor of  . For   the smallest positive integer such that   there exists a computable injective group homomorphism from the subgroup of   of order   to  . If   is small, we can solve DLP in   by using the index calculus attack in  . For arbitrary curves   is very large (around the size of  ); so even though the index calculus attack is quite fast for multiplicative groups of finite fields this attack is not a threat for most curves. The injective function used in this attack is a pairing and there are some applications in cryptography that make use of them. In such applications it is important to balance the hardness of the DLP in   and  ; depending on the security level values of   between 6 and 12 are useful. The subgroup of   is a torus. There exists some independent usage in torus based cryptography.

We also have a problem, if  , the largest prime divisor of the order of the Jacobian, is equal to the characteristic of   By a different injective map we could then consider the DLP in the additive group   instead of DLP on the Jacobian. However, DLP in this additive group is trivial to solve, as can easily be seen. So also these curves, called anomalous curves, are not to be used in DLP.

Order of the JacobianEdit

Hence, in order to choose a good curve and a good underlying finite field, it is important to know the order of the Jacobian. Consider a hyperelliptic curve   of genus   over the field   where   is the power of a prime number and define   as   but now over the field  . It can be shown [6] that the order of the Jacobian of   lies in the interval  , called the Hasse-Weil interval. But there is more, we can compute the order using the zeta-function on hyperelliptic curves. Let   be the number of points on  . Then we define the zeta-function of   as  . For this zeta-function it can be shown [7] that   where   is a polynomial of degree   with coefficients in  . Furthermore   factors as   where   for all  . Here   denotes the complex conjugate of  . Finally we have that the order of   equals  . Hence orders of Jacobians can be found by computing the roots of  .


  1. ^ Déchène, Isabelle (2007). "The Picard Group, or how to build a group from a set" (PDF). Tutorial on Elliptic and Hyperelliptic Curve Cryptography 2007. 
  2. ^ Gaudry, P.; Lubicz, D. (2009). "The arithmetic of characteristic 2 Kummer surfaces and of elliptic Kummer lines". Finite Fields and Their Applications. 15 (2): 246–260. doi:10.1016/j.ffa.2008.12.006. 
  3. ^ Th'eriault, N. (2003). "Index calculus attack for hyperelliptic curves of small genus". Advances in Cryptology - ASIACRYPT 2003. New York: Springer. ISBN 3540406743. 
  4. ^ Enge, Andreas (2002). "Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time". Mathematics of Computation. 71 (238): 729–742. doi:10.1090/S0025-5718-01-01363-1. 
  5. ^ Jasper Scholten and Frederik Vercauteren, An Introduction to Elliptic and Hyperelliptic Curve Cryptography and the NTRU Cryptosystem, section 4
  6. ^ Alfred J. Menezes, Yi-Hong Wu, Robert J. Zuccherato, An elementary introduction to hyperelliptic curves, page 30
  7. ^ Alfred J. Menezes, Yi-Hong Wu, Robert J. Zuccherato, An elementary introduction to hyperelliptic curves, page 29

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